Shadowing and internal chain transitivity (Q2845652)

Let \((X,d)\) be a compact metric space and \(f: X\to X\) a continuous map. The omega-limit set of the orbit \(\{f^n(x)\}_{n=0}^{\infty}\) is the set NEWLINE\[NEWLINE\omega(x) = \bigcap_{m=0}^{\infty} \overline{\{f^n(x): n\geq m\}}.NEWLINE\]NEWLINE The space of omega-limit sets NEWLINE\[NEWLINE\omega(f) = \{A\subset X: \omega(x) = A \text{ for some } x\in X\}NEWLINE\]NEWLINE is the subspace of the hyperspace \(K(X)\) of compacta. The hyperspace \(K(X)\) is endowed with the Hausdorff metric. The task of the paper is to identify conditions for a dynamical system \((X,f)\) under which \(\omega(f)\subset K(X)\) is closed in the Hausdorff topology.NEWLINENEWLINERecall that \(\{x_i\}_{i=0}^{m}\), \(m=0,1,\dots,\infty\), is a \(\delta\)-pseudo-orbit, when \(d(f(x_i),x_{i+1}) < \delta\) for \(i=0,\dots, m-1\) (\(\infty-1=\infty\)). A ``true'' orbit \(\{f^i(z)\}_{i=0}^{m}\), \(z\in X\), \(\epsilon\)-shadows a \(\delta\)-pseudo-orbit \(\{x_i\}_{i=0}^{m}\), when \(d(f^i(z),x_i)<\epsilon\) for all \(i\). The system \((X,f)\) obeys the shadowing property, if for every \(\epsilon >0\) one can find \(\delta > 0\) such that each \(\delta\)-pseudo-orbit is \(\epsilon\)-shadowed by some true orbit. A closed set \(A\subset X\) is internally chain transitive (ICT) for \(f\), whenever for all \(a,b \in A\), \(\delta>0\), there exists a finite \(\delta\)-pseudo-orbit \(\{x_i\}_{i=0}^{m}\) in \(A\) with \(x_0=a\), \(x_m=b\). Analogously to \(\omega(f)\), by \(\mathrm{ICT}(f)\) one denotes the subspace of \(K(X)\) comprising all ICT sets for \(f\).NEWLINENEWLINEFrom the paper we learn that \(\omega(f)\) is closed if and only if \(\omega(f)=\mathrm{ICT}(f)\) under the assumption that \((X,f)\) obeys shadowing. Hence and from the results of other authors it follows that several classes of systems have closed \(\omega(f)\) and \(\mathrm{ICT}(f)=\omega(f)\): (i) interval maps (more generally maps on graphs) with shadowing; (ii) shifts of finite type; (iii) quadratic maps \(f_c(z)= z^2+c\) restricted to its Julia set for certain parameters \(c\in\mathbb{C}\); (iv) topologically hyperbolic maps, in particular topologically Anosov maps etc. Still, it is an open question, whether \(\omega(f)\) is closed, or equivalently (by the main result of the paper) whether \(\omega(f)=\mathrm{ICT}(f)\) for any \((X,f)\) with the shadowing property.